Null Space and Column Space

Null Space

Recall the idea of row reduced Echelon form (RREF) of a matrix!

library(pracma)
#A <- matrix(c(1, -1, 4, 0, 2, 0, -1, 0, -1, -1, 5, 0), nrow=3, ncol=4, byrow=TRUE)
A <- matrix(c(1, -1, 4, 2, 0, -1, -1, -1, 5), nrow=3, ncol=3, byrow=TRUE)
rref(A)
     [,1] [,2] [,3]
[1,]    1    0 -0.5
[2,]    0    1 -4.5
[3,]    0    0  0.0

Question: How to find the null space?

library(MASS)
A <- matrix(c(1, -1, 4, 2, 0, -1, -1, -1, 5), nrow=3, ncol=3, byrow=TRUE)
temp=Null(t(A))
temp/temp[3]
     [,1]
[1,]  0.5
[2,]  4.5
[3,]  1.0
## Example 101
library(pracma)
A <- matrix(c(1, -1, 4, 1, 2, 0, -1, -2, -1, -1, 5, 3),nrow=3, ncol=4, byrow=TRUE)
#A <- matrix(c(1, -1, 4, 1, 0, 2, 0, -1, -2, 0, -1, -1, 5, 3, 0),nrow=3, ncol=5, byrow=TRUE)
rref(A)
     [,1] [,2] [,3] [,4]
[1,]    1    0 -0.5   -1
[2,]    0    1 -4.5   -2
[3,]    0    0  0.0    0
Null(t(A))
           [,1]        [,2]
[1,] -0.3245163  0.49302763
[2,]  0.6935228  0.67894149
[3,]  0.3835873 -0.08774679
[4,] -0.5163100  0.53690102

Question: When do we have a non-unique solution? How to find the complete set of solutions?

See, for example, pp. 218-219 of the book.

Column Space and Row Space

Row Space (A) = Column Space (AT)

Recall the row reduced Echelon form (RREF) of a matrix. How is the function “rref” related to the row space of A?

library(pracma)
A <- matrix(c(1, -1, 4, 2, 0, -1, -1, -1, 5), nrow=3, ncol=3, byrow=TRUE)
#A <- matrix(c(1, -1, 4, 1, 2, 0, -1, -2, -1, -1, 5, 3), nrow=3, ncol=4, byrow=TRUE)
rref(A) # row space
     [,1] [,2] [,3]
[1,]    1    0 -0.5
[2,]    0    1 -4.5
[3,]    0    0  0.0
Null(t(A))
          [,1]
[1,] 0.1078328
[2,] 0.9704950
[3,] 0.2156655
t(rref(t(A))) # column space
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    1   -1    0
## an alternative method
orth(t(A)) # row space (as columns)
           [,1]       [,2]
[1,]  0.0766631 -0.9912088
[2,]  0.2081673  0.1216796
[3,] -0.9750842 -0.0519539
nullspace(A)
          [,1]
[1,] 0.1078328
[2,] 0.9704950
[3,] 0.2156655
orth(A) # column space
           [,1]       [,2]
[1,] -0.6067397 -0.5463823
[2,]  0.1698111 -0.7986431
[3,] -0.7765508  0.2522608

Exercise

  1. Use the rref function
  2. Use the Null function
  3. Use the null space function